Everywhere divergence of one-sided ergodic Hilbert transform
| Autor: | Jörg Schmeling, Ai-Hua Fan |
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| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Series (mathematics) Continuous function (set theory) Diophantine equation 010102 general mathematics Function (mathematics) 01 natural sciences Irrational rotation 010104 statistics & probability symbols.namesake symbols Ergodic theory Geometry and Topology Hilbert transform 0101 mathematics Fourier series Mathematics |
| Zdroj: | Annales de l'Institut Fourier. 68:2477-2500 |
| ISSN: | 1777-5310 |
| DOI: | 10.5802/aif.3214 |
| Popis: | For a given number α ϵ (0, 1) and a 1-periodic function f, we study the convergence of the series Σ∞ n=1f(x+nα)/n, called one-sided Hilbert transform relative to the rotation x → x + α mod 1. Among others, we prove that for any non-polynomial function of class C2 having Taylor-Fourier series (i.e. Fourier coefficients vanish on ℤ-), there exists an irrational number α (actually a residual set of α) such that the series diverges for all x. We also prove that for any irrational number α, there exists a continuous function f such that the series diverges for all x. The convergence of general series Σ∞ n=1 anf(x + nα) is also discussed in different cases involving the diophantine property of the number α and the regularity of the function f. |
| Databáze: | OpenAIRE |
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